Image Processing Reference

In-Depth Information

20

Nonlin. Local Models

Global Model

Eq. Constraints

Ineq. Constraints

20

Nonlin. Local Models

Global Model

Eq. Constraints

Ineq. Constraints

15

15

10

10

5

5

0

0

20

40

60

80

100

20

40

60

80

100

Frame #

Frame #

Figure 4.16:
Mean vertex-to-vertex distance between reconstructions and ground-truth meshes. The

reconstructions were obtained from synthetic correspondence (left) and SIFT correspondences (right).

Results were obtained with the linear local models of
Salzmann and Fua
[
2011
] with distance equalities

and inequalities, as well as with the nonlinear local models of
Salzmann
et al.
[
2008b
] (green) and the

global models of
Salzmann
et al.
[
2008a
] with distance equalities (cyan). The largest deformation appears

around frame 60, where the difference in accuracy is the greatest.

formulated as an SOCP problem by introducing slack variables
Boyd and Vandenberghe
[
2004
].

Doing so allows to re-write the problem as

c
+
r
−
λ
d
x
T
d

minimize

X
,
c
,
r

(4.17)

subject to

Mx

2
≤

c
,

(
x

−

x
0
)
2
≤
r
,

∀
(j, k)
∈
E

v
k
−

v
j
≤
l
j,k
,

,

which, like the problem of Eq.
4.4
, can be solved using a standard solver
Sturm
[
1999
]. As depicted by

Fig.
4.16
, the resulting solution of
Salzmann and Fua
[
2011
] tends to outperform the global smooth-

ness methods of
Salzmann
et al.
[
2008a
], as well as the nonlinear local models of
Salzmann
et al.

[
2008b
].

While the optimization problem of Eq.
4.16
is convex, it also is very large and takes a long

time to solve. As a result, this approach is ill-suited to real-time applications. One way to alleviate

this problem is to leverage the availability of a good initial solution to exploit efficient least-squares

resolution techniques. To this end, the problem of Eq.
4.16
can be reformulated as the constrained

least-squares minimization problem

2

2

minimize

X

Mx

+

(
x

−

x
0
)

(4.18)

subject to

v
k
−

v
j
=
l
j,k
,

∀
(j, k)
∈
E

.

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